TABLE OF CONTENTS Standard finite element methods have been shown to be capable, in principle, of dealing with any two- or three- (or even four-) [*] dimensional situations. Nevertheless, the cost of solutions increases greatly with each dimension added and indeed, on occasion, overtaxes the available computer capability. It is therefore always desirable to search for alternatives that may reduce computational effort. One such class of processes of quite wide applicability will be illustrated in this chapter.
In many physical problems the situation is such that the geometry and material properties do not vary along one coordinate direction. However, the 'load' terms may still exhibit a variation in that direction, preventing the use of such simplifying assumptions as those that, for instance, permitted a two-dimensional plane strain or axisymmetric analysis to be substituted for a full three-dimensional treatment. In such cases it is possible still to consider a 'substitute' problem, not involving the particular coordinate (along which the geometry and properties do not vary), and to synthesize the true answer from a series of such simplified solutions.
The method to be described is of quite general use and, obviously, is not limited to structural situations. It will be convenient, however, to use the nomenclature of structural mechanics and to use potential energy minimization as an example.
We shall confine our attention to problems of minimizing a quadratic functional for a linear elastic material. The interpretation of the process involved as the application of partial discretization leading to semi-discrete...
